Spatial analysis of Kensington and Chelsea district in London (on OA level) - including house prices.

Model part

Finding neighbours:

Global Spatial Autocorrelation

Moran I Test:

## 
##  Moran I test under randomisation
## 
## data:  OA.Census$employed  
## weights: listw    
## 
## Moran I statistic standard deviate = 14.574, p-value < 2.2e-16
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##      0.3543126571     -0.0015873016      0.0005963801

Employed has 0.34 moran statistic so it has a slight postitive autocorrelation - we may say that the data does spatially cluster.

Local Spatial Autocorrelation

LISA cluster map

GETIS-ORD

## Variable(s) "gstat" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.

Linear model - employed

Using backward variables selection we eliminated most of the insignificant variables and came up with following model:

## 
## Call:
## lm(formula = OA.Census$employed ~ . - 1, data = OA.Census[, sig_cols_2])
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -28.3137  -3.2513   0.3911   3.7125  19.6324 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## white          0.31933    0.02140  14.923  < 2e-16 ***
## black_african  0.28316    0.05154   5.494 5.71e-08 ***
## single         0.12605    0.01913   6.588 9.44e-11 ***
## lowest_quali   0.58775    0.10107   5.815 9.65e-09 ***
## highest_quali  0.54895    0.02709  20.267  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.733 on 626 degrees of freedom
## Multiple R-squared:  0.992,  Adjusted R-squared:  0.9919 
## F-statistic: 1.548e+04 on 5 and 626 DF,  p-value: < 2.2e-16

GWR - employed

## Adaptive q: 0.381966 CV score: 20668.69 
## Adaptive q: 0.618034 CV score: 20832.37 
## Adaptive q: 0.236068 CV score: 20444.63 
## Adaptive q: 0.145898 CV score: 20428.7 
## Adaptive q: 0.1756323 CV score: 20424.39 
## Adaptive q: 0.1743792 CV score: 20424.91 
## Adaptive q: 0.1987167 CV score: 20425.05 
## Adaptive q: 0.1863925 CV score: 20421.06 
## Adaptive q: 0.1910999 CV score: 20420.96 
## Adaptive q: 0.1893096 CV score: 20419.23 
## Adaptive q: 0.1887771 CV score: 20419.24 
## Adaptive q: 0.1890814 CV score: 20419.23 
## Adaptive q: 0.1899934 CV score: 20419.2 
## Adaptive q: 0.1904161 CV score: 20419.66 
## Adaptive q: 0.1897322 CV score: 20419.21 
## Adaptive q: 0.1901549 CV score: 20419.2 
## Adaptive q: 0.1902546 CV score: 20419.35 
## Adaptive q: 0.1900932 CV score: 20419.2 
## Adaptive q: 0.1901956 CV score: 20419.24 
## Adaptive q: 0.1901549 CV score: 20419.2
## Call:
## gwr(formula = OA.Census$employed ~ . - 1, data = OA.Census[, 
##     sig_cols_2], adapt = GWRbandwidth, hatmatrix = TRUE, se.fit = TRUE)
## Kernel function: gwr.Gauss 
## Adaptive quantile: 0.1901549 (about 119 of 631 data points)
## Summary of GWR coefficient estimates at data points:
##                   Min.  1st Qu.   Median  3rd Qu.     Max. Global
## white         0.270580 0.288871 0.299836 0.313213 0.335517 0.3193
## black_african 0.023445 0.195534 0.242522 0.281999 0.351382 0.2832
## single        0.027817 0.052734 0.085185 0.213740 0.313398 0.1260
## lowest_quali  0.207876 0.276701 0.792704 0.904568 1.053125 0.5878
## highest_quali 0.462734 0.496910 0.596113 0.614051 0.651189 0.5490
## Number of data points: 631 
## Effective number of parameters (residual: 2traceS - traceS'S): 22.44898 
## Effective degrees of freedom (residual: 2traceS - traceS'S): 608.551 
## Sigma (residual: 2traceS - traceS'S): 5.564811 
## Effective number of parameters (model: traceS): 16.77853 
## Effective degrees of freedom (model: traceS): 614.2215 
## Sigma (model: traceS): 5.539064 
## Sigma (ML): 5.464925 
## AICc (GWR p. 61, eq 2.33; p. 96, eq. 4.21): 3970.666 
## AIC (GWR p. 96, eq. 4.22): 3950.797 
## Residual sum of squares: 18845.07 
## Quasi-global R2: 0.6938792

LINEAR MODEL - PRICES

## 
## Call:
## lm(formula = OA.Census.mp$mean_price ~ ., data = OA.Census.mp[sig_cols])
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2221130  -437500  -119641   227375  6426924 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    6183339     735000   8.413 3.64e-16 ***
## single          -12374       4913  -2.519  0.01207 *  
## muslim          -17097       7559  -2.262  0.02411 *  
## highest_quali    12094       5790   2.089  0.03721 *  
## jewish           70316      23129   3.040  0.00248 ** 
## asian            -7480       9048  -0.827  0.40879    
## one_car         -38962       9201  -4.234 2.70e-05 ***
## no_cars         -44399       7206  -6.161 1.42e-09 ***
## Age_30_44       -12502       9360  -1.336  0.18220    
## employed        -15271       7348  -2.078  0.03817 *  
## private_rent      5909       3734   1.582  0.11415    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 860500 on 537 degrees of freedom
## Multiple R-squared:  0.3998, Adjusted R-squared:  0.3886 
## F-statistic: 35.76 on 10 and 537 DF,  p-value: < 2.2e-16

GWR - PRICES

## Adaptive q: 0.381966 CV score: 4.210973e+14 
## Adaptive q: 0.618034 CV score: 4.182034e+14 
## Adaptive q: 0.763932 CV score: 4.172618e+14 
## Adaptive q: 0.9032981 CV score: 4.168388e+14 
## Adaptive q: 0.9602445 CV score: 4.167327e+14 
## Adaptive q: 0.9384929 CV score: 4.167674e+14 
## Adaptive q: 0.9754298 CV score: 4.16696e+14 
## Adaptive q: 0.9848148 CV score: 4.16671e+14 
## Adaptive q: 0.990615 CV score: 4.166489e+14 
## Adaptive q: 0.9941998 CV score: 4.166265e+14 
## Adaptive q: 0.9964153 CV score: 4.166109e+14 
## Adaptive q: 0.9977845 CV score: 4.165974e+14 
## Adaptive q: 0.9986307 CV score: 4.165924e+14 
## Adaptive q: 0.9991538 CV score: 4.16591e+14 
## Adaptive q: 0.9995096 CV score: 4.1659e+14 
## Adaptive q: 0.9996969 CV score: 4.165894e+14 
## Adaptive q: 0.9998127 CV score: 4.165891e+14 
## Adaptive q: 0.9998842 CV score: 4.165889e+14 
## Adaptive q: 0.9999285 CV score: 4.165888e+14 
## Adaptive q: 0.9999285 CV score: 4.165888e+14
## Call:
## gwr(formula = OA.Census.mp$mean_price ~ ., data = OA.Census.mp[, 
##     sig_cols], adapt = GWRbandwidth, hatmatrix = TRUE, se.fit = TRUE)
## Kernel function: gwr.Gauss 
## Adaptive quantile: 0.9999285 (about 547 of 548 data points)
## Summary of GWR coefficient estimates at data points:
##                    Min.   1st Qu.    Median   3rd Qu.      Max.    Global
## X.Intercept.  6088655.0 6120270.2 6166817.8 6319731.2 6344257.2 6183339.3
## single         -12875.8  -12827.6  -12476.1  -12242.1  -12106.5  -12373.7
## muslim         -18901.7  -18208.7  -17912.6  -17555.4  -17225.3  -17096.8
## highest_quali   11117.2   11480.6   11588.8   11737.1   12170.0   12094.0
## jewish          65184.9   66420.8   67963.5   70085.6   71072.1   70316.2
## asian           -7583.5   -7500.8   -7284.7   -7204.3   -7159.6   -7479.5
## one_car        -40864.5  -40397.0  -38314.1  -37765.0  -37536.6  -38961.7
## no_cars        -45261.9  -44957.5  -44167.7  -43936.7  -43719.8  -44399.1
## Age_30_44      -14652.5  -13714.0  -13312.8  -12977.9  -12410.5  -12501.7
## employed       -15235.4  -14847.0  -14650.6  -14381.1  -13797.6  -15271.2
## private_rent     5505.2    5679.7    6169.0    6293.5    6343.6    5908.7
## Number of data points: 548 
## Effective number of parameters (residual: 2traceS - traceS'S): 13.47969 
## Effective degrees of freedom (residual: 2traceS - traceS'S): 534.5203 
## Sigma (residual: 2traceS - traceS'S): 860638.4 
## Effective number of parameters (model: traceS): 12.30621 
## Effective degrees of freedom (model: traceS): 535.6938 
## Sigma (model: traceS): 859695.2 
## Sigma (ML): 849987.5 
## AICc (GWR p. 61, eq 2.33; p. 96, eq. 4.21): 16546.15 
## AIC (GWR p. 96, eq. 4.22): 16531.13 
## Residual sum of squares: 3.959183e+14 
## Quasi-global R2: 0.402407

Interpolation

## 
##      PLEASE NOTE:  The components "delsgs" and "summary" of the
##  object returned by deldir() are now DATA FRAMES rather than
##  matrices (as they were prior to release 0.0-18).
##  See help("deldir").
##  
##      PLEASE NOTE: The process that deldir() uses for determining
##  duplicated points has changed from that used in version
##  0.0-9 of this package (and previously). See help("deldir").

Inverse distance weighting

## [inverse distance weighted interpolation]

3D PLOTTING

Geostatistical interpolation